Calculate the force of magnetic attraction between the northern and southern hemispheres of a uniformly charged spinning spherical shell, with radius R, angular velocity ω and surface charge density σ.
Maxwell Tensor : Tm = [/B](1/μ) * ((B*n)B - (1/2)*B2 n)
B_in = (2/3)*μσRω*z
B_out = μm/(4πr3) * (2cosθ r + sinθ θ)
where m = (4/3)*πR3(σωR)
The Attempt at a Solution
I do see in my solution that :
1-We use a surface consisting of the entire equatorial plane, CLOSING IT with a hemispherical surface at infinity where (since the field is zero out there) contribution is zero.
for r>R :
B = μm/(4πr3) θ = - μm/(4πr3) z (Why ?! )
2- We have a equatorial circular disk. We use B inside and da = rdrdφ. Direction is in z-axis!
I don't want the solution because I already have it but MY QUESTIONS are :
A) How should think and imagine this spheres ?!
Should I think like this and use those formulas :
B) Why we do write B = μm/(4πr3) θ = - μm/(4πr3) z
Why is θ =-z in this case ?! I don't get it !
I know the rest. That is just integrating